Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems are secure assuming that it is difficult to factor a large integer composed of two or more large prime factors. For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible: this is the "elliptic curve discrete logarithm problem" or ECDLP. The entire security of ECC depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original and product points. The size of the elliptic curve determines the difficulty of the problem.

The primary benefit promised by ECC is a smaller key size, reducing storage and transmission requirements, i.e. that an elliptic curve group could provide the same level of security afforded by an RSA-based system with a large modulus and correspondingly larger key: for example, a 256-bit ECC public key should provide comparable security to a 3072-bit RSA public key.

For current cryptographic purposes, an elliptic curve is a plane curve over a finite field (rather than the real numbers) which consists of the points satisfying the equation

along with a distinguished point at infinity, denoted ∞. (The coordinates here are to be chosen from a fixed finite field of characteristic not equal to 2 or 3, or the curve equation will be somewhat more complicated.)

At the RSA Conference 2005, the National Security Agency (NSA) announced Suite B which exclusively uses ECC for digital signature generation and key exchange. The suite is intended to protect both classified and unclassified national security systems and information.[6]

To use ECC all parties must agree on all the elements defining the elliptic curve, that is, the domain parameters of the scheme. The field is defined by p in the prime case and the pair of m and f in the binary case. The elliptic curve is defined by the constants a and b used in its defining equation. Finally, the cyclic subgroup is defined by its generator (a.k.a. base point) G. For cryptographic application the order of G, that is the smallest positive number n such that , is normally prime. Since n is the size of a subgroup of it follows from Lagrange's theorem that the number is an integer. In cryptographic applications this number h, called the cofactor, must be small () and, preferably, . Let us summarize: in the prime case the domain parameters are and in the binary case they are .

Unless there is an assurance that domain parameters were generated by a party trusted with respect to their use, the domain parameters must be validated before use.

The generation of domain parameters is not usually done by each participant since this involves computing the number of points on a curve which is time-consuming and troublesome to implement. As a result several standard bodies published domain parameters of elliptic curves for several common field sizes. Such domain parameters are commonly known as "standard curves" or "named curves"; a named curve can be referenced either by name or by the unique object identifier defined in the standard documents:

SECG test vectors are also available.[7] NIST has approved many SECG curves, so there is a significant overlap between the specifications published by NIST and SECG. EC domain parameters may be either specified by value or by name.

If one (despite the above) wants to construct one's own domain parameters, one should select the underlying field and then use one of the following strategies to find a curve with appropriate (i.e., near prime) number of points using one of the following methods:

Curves such that n divides (where p is the characteristic of the field – q for a prime field, or for a binary field) for sufficiently small B are vulnerable to Menezes–Okamoto–Vanstone (MOV) attack[11][12] which applies usual Discrete Logarithm Problem (DLP) in a small degree extension field of to solve ECDLP. The bound B should be chosen so that discrete logarithms in the field are at least as difficult to compute as discrete logs on the elliptic curve .[13]

Curves such that are vulnerable to the attack that maps the points on the curve to the additive group of [14][15][16]

Since all the fastest known algorithms that allow one to solve the ECDLP (baby-step giant-step, Pollard's rho, etc.), need steps, it follows that the size of the underlying field should be roughly twice the security parameter. For example, for 128-bit security one needs a curve over , where . This can be contrasted with finite-field cryptography (e.g., DSA) which requires[17] 3072-bit public keys and 256-bit private keys, and integer factorization cryptography (e.g., RSA) which requires a 3072-bit value of n, where the private key should be just as large. However the public key may be smaller to accommodate efficient encryption, especially when processing power is limited.

The hardest ECC scheme (publicly) broken to date had a 112-bit key for the prime field case and a 109-bit key for the binary field case. For the prime field case this was broken in July 2009 using a cluster of over 200 PlayStation 3 game consoles and could have been finished in 3.5 months using this cluster when running continuously (see [18]). For the binary field case, it was broken in April 2004 using 2600 computers for 17 months.[19]

A current project is aiming at breaking the ECC2K-130 challenge by Certicom, by using a wide range of different hardware: CPUs, GPUs, FPGA.[20]

A close examination of the addition rules shows that in order to add two points one needs not only several additions and multiplications in but also an inversion operation. The inversion (for given find such that ) is one to two orders of magnitude slower[21] than multiplication. Fortunately, points on a curve can be represented in different coordinate systems which do not require an inversion operation to add two points. Several such systems were proposed: in the projective system each point is represented by three coordinates using the following relation: , ; in the Jacobian system a point is also represented with three coordinates , but a different relation is used: , ; in the López–Dahab system the relation is , ; in the modified Jacobian system the same relations are used but four coordinates are stored and used for calculations ; and in the Chudnovsky Jacobian system five coordinates are used . Note that there may be different naming conventions, for example, IEEE P1363-2000 standard uses "projective coordinates" to refer to what is commonly called Jacobian coordinates. An additional speed-up is possible if mixed coordinates are used.[22]

Reduction modulo p (which is needed for addition and multiplication) can be executed much faster if the prime p is a pseudo-Mersenne prime that is , for example, or Compared to Barrett reduction there can be an order of magnitude speedup.[23] The speedup here is a practical rather than theoretical one, and derives from the fact that the moduli of numbers against numbers near powers of two can be performed efficiently by computers operating on binary numbers with bitwise operations.

The curves over with pseudo-Mersenne p are recommended by NIST. Yet another advantage of the NIST curves is the fact that they use a = −3, which improves addition in Jacobian coordinates.

Many of the efficiency-related decisions in NIST FIPS 186-2 are sub-optimal. Other curves are more secure and run just as fast.[24]

Five prime fields for certain primes p of sizes 192, 224, 256, 384, and 521[25] bits. For each of the prime fields, one elliptic curve is recommended.

Five binary fields for m equal 163, 233, 283, 409, and 571. For each of the binary fields, one elliptic curve and one Koblitz curve was selected.

The NIST recommendation thus contains a total of five prime curves and ten binary curves. The curves were ostensibly chosen for optimal security and implementation efficiency.[26]

In 2013, the New York Times stated that Dual Elliptic Curve Deterministic Random Bit Generation (or Dual_EC_DRBG) had been included as a NIST national standard due to the influence of NSA, which had included a deliberate weakness in the algorithm and the recommended elliptic curve. RSA Security in September 2013 issued an advisory recommending that its customers discontinue using any software based on Dual_EC_DRBG.[27][28] In the wake of the exposure of Dual_EC_DRBG as "an NSA undercover operation", cryptography experts have also expressed concern over the security of the NIST recommended elliptic curves[29], suggesting a return to encryption based on non-elliptic-curve groups.

Unlike most other DLP systems (where it is possible to use the same procedure for squaring and multiplication) the EC addition is significantly different for doubling () and general addition () depending on the coordinate system used. Consequently, it is important to counteract side channel attacks (e.g., timing or simple/differential power analysis attacks) using, for example, fixed pattern window (a.k.a. comb) methods[clarification needed][30] (note that this does not increase the computation time). Another concern for ECC-systems is the danger of fault attacks, especially when running on smart cards.[31]

Cryptographic experts have also expressed concerns that the National Security Agency has inserted a backdoor into at least one elliptic curve-based pseudo random generator.[32] One analysis of the possible backdoor concluded that an adversary in possession of the algorithm's secret key could obtain encryption keys given only 32 bytes of ciphertext.[33]

Elliptic curve cryptography is vulnerable to a modified Shor's algorithm for solving the discrete logarithm problem on elliptic curves.[34] A quantum computer to attack elliptic curve cryptography can be less than half the size of a quantum computer to break an equivalently classically secure version of RSA. This is due to the fact that smaller key sizes of elliptic curves are needed to match the classical security of RSA. The work of Proos and Zalka show how a quantum computer to break 2048-bit RSA requires roughly 4096 qubits while a quantum computer to break the equivalently secure 224-bit Elliptic Curve Cryptography requires between 1300 and 1600 qubits. Depending on the growth rate of quantum computers in the future, elliptic curve cryptosystems may become attackable by a quantum computer many years before an equivalently secure RSA scheme.[35]

To avoid quantum computing concerns, an elliptic curve based alternative to Elliptic Curve Diffie Hellman which is not susceptible to Shor's attack is the Supersingular Isogeny Diffie–Hellman Key Exchange of De Feo, Jao and Plut. They use elliptic curve isogenies to create a drop in replacement for the quantum attackable Diffie–Hellman and Elliptic curve Diffie–Hellman key exchanges. This key exchange utilizes the same elliptic curve computational primitives of existing elliptic curve cryptography and requires computational and transmission overhead similar to many currently used public key systems.[36]

^Semaev, I. (1998). "Evaluation of discrete logarithm in a group of p-torsion points of an elliptic curve in characteristic p". Mathematics of Computation67 (221): 353–356. doi:10.1090/S0025-5718-98-00887-4.

^Kim Zetter, RSA Tells Its Developer Customers: Stop Using NSA-Linked AlgorithmWired, 19 September 2013. "Recommending against the use of SP 800-90A Dual Elliptic Curve Deterministic Random Bit Generation: NIST strongly recommends that, pending the resolution of the security concerns and the re-issuance of SP 800-90A, the Dual_EC_DRBG, as specified in the January 2012 version of SP 800-90A, no longer be used."